Actuator device and method for manufacturing actuator device

ABSTRACT

An actuator device includes a support portion; a first movable portion; a second movable portion; a first connection portion that connects the first and second movable portions such that the first movable portion is swingable around a first axis; a second connection portion that connects the second movable portion and the support portion such that the first movable portion is swingable around the first axis. Two natural angular frequencies ω1 and ω2 (where ω1&lt;ω2) for vibration of the first and second movable portions around the first axis satisfy one of the following equation (1) and equation (2) and do not satisfy the other,[Equation⁢⁢1]0&lt;1-(ω1ωii)2≤0.2(1)[Equation⁢⁢2]0&lt;(ω2ωii)2-1≤0.2(2)In the above equations, ωii=(ki/ji)1/2, ki is a torsional spring constant of the first connection portion around the first axis, and ji is an inertia moment of the first movable portion around the first axis.

TECHNICAL FIELD

One aspect of the present disclosure relates to an actuator device and amethod for producing an actuator device.

BACKGROUND ART

As a microelectromechanical systems (MEMS) device, there is known anactuator device including a support portion, a first movable portion, asecond movable portion that surrounds the first movable portion, a pairof first connection portions that connect the first movable portion andthe second movable portion to each other on a first axis, and a pair ofsecond connection portions that connect the second movable portion andthe support portion to each other on the first axis (for example, referto Patent Literature 1). In the actuator device described in PatentLiterature 1, the first movable portion is provided with a lightreflection portion, and when an AC voltage is applied to the lightreflection portion, the second movable portion rotates around the firstaxis and accordingly, the first movable portion rotates around the firstaxis.

CITATION LIST Patent Literature

Patent Literature 1: Japanese Unexamined Patent Publication No.2005-99760

SUMMARY OF INVENTION Technical Problem

In the actuator device as described above, from the viewpoint of highefficiency, it is preferable that a ratio of the deflection angle of thefirst movable portion to the deflection angle of the second movableportion is large. In addition, the actuator device as described aboverequires a stable operation.

One aspect of the present disclosure is to provide an actuator devicecapable of realizing a highly efficient and stable operation, and amethod for producing such an actuator device.

Solution to Problem

According to one aspect of the present disclosure, there is provided anactuator device including: a support portion; a first movable portion; asecond movable portion that has a frame shape and surrounds the firstmovable portion; a first connection portion that connects the firstmovable portion and the second movable portion to each other such thatthe first movable portion is swingable around a first axis; a secondconnection portion that connects the second movable portion and thesupport portion to each other such that the first movable portion isswingable around the first axis by vibrating the second movable portion;and a drive unit that applies a drive force to the second movableportion. Two natural angular frequencies ω₁ and ω₂ (where ω₁<ω₂) forvibration of the first movable portion and the second movable portionaround the first axis satisfy one of the following first equation (1)and second equation (2) and do not satisfy the other.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 1} \right\rbrack & \; \\{0 < {1 - \left( \frac{\omega_{1}}{\omega_{ii}} \right)^{2}} \leq 0.2} & (1) \\\left\lbrack {{Equation}\mspace{14mu} 2} \right\rbrack & \; \\{0 < {\left( \frac{\omega_{2}}{\omega_{ii}} \right)^{2} - 1} \leq 0.2} & (2)\end{matrix}$

In the above equations, ω_(ii)=(k_(i)/j_(i))^(1/2), k_(i) is a torsionalspring constant of the first connection portion around the first axis,and j_(i) is an inertia moment of the first movable portion around thefirst axis.

According to one aspect of the present disclosure, there is provided anactuator device including: a support portion; a first movable portion; asecond movable portion that has a frame shape and surrounds the firstmovable portion; a first connection portion that connects the firstmovable portion and the second movable portion to each other such thatthe first movable portion is swingable around a first axis; a secondconnection portion that connects the second movable portion and thesupport portion to each other; and a drive unit that is provided in atleast one of the first movable portion and the first connection portionto apply a drive force to the first movable portion. Two natural angularfrequencies ω₁ and ω₂ (where ω₁<ω₂) for vibration of the first movableportion and the second movable portion around the first axis satisfy oneof the following first equation (3) and second equation (4) and do notsatisfy the other.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 3} \right\rbrack & \; \\{0 < \frac{4\omega_{io}^{2}}{\omega_{oo}^{2} - \omega_{1}^{2}} \leq 1} & (3) \\\left\lbrack {{Equation}\mspace{14mu} 4} \right\rbrack & \; \\{\frac{\omega_{oo}^{2} - \omega_{2}^{2}}{\omega_{io}^{2}} \leq {- 6}} & (4)\end{matrix}$

In the above equations, ω_(io)=(k_(i)/j_(o))^(1/2),ω_(oo)=(k_(o)/j_(o))^(1/2), k_(i) is a torsional spring constant of thefirst connection portion around the first axis, k_(o) is a torsionalspring constant of the second connection portion around the first axis,and j_(o) is an inertia moment of the second movable portion around thefirst axis.

In these actuator devices, when the two natural angular frequencies ω₁and ω₂ satisfy the first equation and do not satisfy the secondequation, the first movable portion and the second movable portion swingat the natural angular frequency ω₁, so that the ratio of the deflectionangle of the first movable portion to the deflection angle of the secondmovable portion (hereinafter, referred to as an “amplitude ratio”) canbe 5 or more. Meanwhile, when the two natural angular frequencies ω₁ andω₂ satisfy the second equation and do not satisfy the first equation,the first movable portion and the second movable portion swing at thenatural angular frequency ω₂, so that the amplitude ratio can be 5 ormore. Therefore, in both cases, the amplitude ratio can be greatlysecured, and the efficiency of the operation can be improved.Furthermore, since only one of the first equation and the secondequation is satisfied and the other is not satisfied, the differencebetween the natural angular frequencies ω₁ and ω₂ can be greatlysecured, and an unnecessary resonance response can be suppressed tostabilize the operation. As a result, according to these actuatordevices, a highly efficient and stable operation can be implemented.

The second connection portion may connect the second movable portion andthe support portion to each other such that the second movable portionis swingable around a second axis intersecting the first axis. In thiscase, the second movable portion is swingable around the second axis,together with the first movable portion.

The two natural angular frequencies may satisfy the first equation andmay not satisfy the second equation. A torsional spring constant of thesecond connection portion around the first axis may be larger than thetorsional spring constant of the first connection portion around thefirst axis. In this case, the amplitude ratio can be more greatlysecured.

A total length of the second connection portion may be shorter than atotal length of the first connection portion. In this case, thetorsional spring constant of the second connection portion can besuitably made larger than the torsional spring constant of the firstconnection portion. Furthermore, instead of increasing thecross-sectional areas of the second connection portion, the length ofthe second connection portion is shortened to increase the torsionalspring constant of the second connection portion; and thereby, while thesize of the actuator device is reduced, the amplitude ratio can begreatly secured.

The second connection portion may include a plurality of members, eachof which connects the second movable portion and the support portion toeach other. The plurality of members may be disposed side by side alonga direction intersecting the first axis. In this case, the torsionalspring constant of the second connection portion can be more suitablymade larger than the torsional spring constant of the first connectionportion.

The plurality of members may include a first member disposed on thefirst axis, and a pair of second members disposed to interpose the firstmember between the second members. In this case, the torsional springconstant of the second connection portion can be more suitably madelarger than the torsional spring constant of the first connectionportion.

The actuator device according to one aspect of the present disclosuremay further include a wiring extending from the second movable portionto the support portion via the second connection portion. The wiring maybe disposed to pass through the first member. In this case, since thewiring is disposed to pass through the first member which is subjectedto a smaller stress than the second member during vibration of thesecond movable portion, deterioration of the wiring in the secondconnection portion can be suppressed.

The two natural angular frequencies may satisfy the second equation andmay not satisfy the first equation. A torsional spring constant of thesecond connection portion around the first axis may be smaller than thetorsional spring constant of the first connection portion around thefirst axis. In this case, the amplitude ratio can be more greatlysecured.

A total length of the second connection portion may be longer than atotal length of the first connection portion. In this case, thetorsional spring constant of the second connection portion can besuitably made smaller than the torsional spring constant of the firstconnection portion.

The second connection portion may extend in a meandering manner. In thiscase, the torsional spring constant of the second connection portion canbe more suitably made smaller than the torsional spring constant of thefirst connection portion. Furthermore, since the second connectionportion extends in a meandering manner, impact resistance can be furtherimproved, for example, as compared to when the second connection portionextends linearly.

The support portion, the first movable portion, the second movableportion, the first connection portion, and the second connection portionmay be configured by a semiconductor substrate. Even in the actuatordevice configured in such a manner, a highly efficient and stableoperation can be implemented.

The actuator device according to one aspect of the present disclosuremay further include a sensing coil provided in the first movableportion; and a wiring that is connected to the sensing coil and extendsto the second movable portion via the first connection portion. Thefirst connection portion may be made of a semiconductor material. Aportion of the wiring, the portion being located on the first connectionportion, may be configured by a diffusion region where impurities arediffused in the semiconductor material. In this case, a relatively largestress is applied to the portion of the wiring during swinging of thefirst movable portion, the portion being located on the first connectionportion; however, since the portion is configured by the diffusionregion, deterioration of the wiring in the first connection portion canbe suppressed.

The diffusion region may extend from the first connection portion to thefirst movable portion and the second movable portion. A width of thediffusion region in the first movable portion and the second movableportion may be wider than a width of the diffusion region in the firstconnection portion. In this case, the resistance of the wiring can bereduced.

A portion of the wiring, the portion being located on the first movableportion, and a portion of the wiring, the portion being located on thesecond movable portion, may be made of a metallic material. A width of acontact portion between the diffusion region and each of the portionlocated on the first movable portion and the portion located on thesecond movable portion may be wider than a width of the first connectionportion. In this case, the resistance of the wiring can be furtherreduced. In addition, even when a misalignment occurs during productionor the like, the diffusion region can be reliably electrically connectedto each of the portion located on the first movable portion and theportion located on the second movable portion.

According to one aspect of the present disclosure, there is provided amethod for producing an actuator device including a support portion, afirst movable portion, a second movable portion that has a frame shapeand surrounds the first movable portion, a first connection portion thatconnects the first movable portion and the second movable portion toeach other such that the first movable portion is swingable around afirst axis, a second connection portion that connects the second movableportion and the support portion to each other such that the firstmovable portion is swingable around the first axis by vibrating thesecond movable portion, and a drive unit that applies a drive force tothe second movable portion, in which two natural angular frequencies ω₁and ω₂ (where ω₁<ω₂) for vibration of the first movable portion and thesecond movable portion around the first axis satisfy one of thefollowing equation (5) and equation (6) and do not satisfy the other.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 5} \right\rbrack & \; \\{0 < {1 - \left( \frac{\omega_{1}}{\omega_{ii}} \right)^{2}} \leq 0.2} & (5) \\\left\lbrack {{Equation}\mspace{14mu} 6} \right\rbrack & \; \\{0 < {\left( \frac{\omega_{2}}{\omega_{ii}} \right)^{2} - 1} \leq 0.2} & (6)\end{matrix}$

In the above equations, ω_(ii)=(k_(i)/j_(i))^(1/2) k_(i) is a torsionalspring constant of the first connection portion around the first axis,and j_(i) is an inertia moment of the first movable portion around thefirst axis.

According to one aspect of the present disclosure, there is provided amethod for producing an actuator device including a support portion, afirst movable portion, a second movable portion that has a frame shapeand surrounds the first movable portion, a first connection portion thatconnects the first movable portion and the second movable portion toeach other such that the first movable portion is swingable around afirst axis, a second connection portion that connects the second movableportion and the support portion to each other, and a drive unit that isprovided in at least one of the first movable portion and the firstconnection portion to apply a drive force to the first movable portion,in which two natural angular frequencies ω₁ and ω₂ (where ω₁<ω₂) forvibration of the first movable portion and the second movable portionaround the first axis satisfy one of the following equation (7) andequation (8) and do not satisfy the other.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 7} \right\rbrack & \; \\{0 < \frac{4\omega_{io}^{2}}{\omega_{oo}^{2} - \omega_{1}^{2}} \leq 1} & (7) \\\left\lbrack {{Equation}\mspace{14mu} 8} \right\rbrack & \; \\{\frac{\omega_{oo}^{2} - \omega_{2}^{2}}{\omega_{io}^{2}} \leq {- 6}} & (8)\end{matrix}$

In the above equations, ω_(io)=(k_(i)/j_(o))^(1/2),ω_(oo)=(k_(o)/j_(o))^(1/2), k_(i) is a torsional spring constant of thefirst connection portion around the first axis, k_(o) is a torsionalspring constant of the second connection portion around the first axis,and j_(o) is an inertia moment of the second movable portion around thefirst axis.

According to the actuator device produced by the production methods, ahighly efficient and stable operation can be implemented for the reasonsdescribed above.

Advantageous Effects of Invention

According to one aspect of the present disclosure, it is possible toprovide the actuator device capable of realizing a highly efficient andstable operation, and the method for producing such an actuator device.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 is a plan view of a mirror device according to a firstembodiment.

FIG. 2 is a plan view of a mirror device according to a secondembodiment.

FIG. 3 is a partial enlarged view of FIG. 2.

FIG. 4 is a cross-sectional view along line IV-IV of FIG. 3.

FIGS. 5(a) and 5(b) are graphs showing examples of a relationshipbetween the input angular frequency and the deflection angle.

FIG. 6(a) is a schematic view illustrating an example of a relationshipbetween the deflection angles of a first movable portion and a secondmovable portion at a resonance point A of FIG. 5(a), and FIG. 6(b) is aschematic view illustrating an example of a relationship between thedeflection angles of the first movable portion and the second movableportion at a resonance point B of FIG. 5(a).

FIG. 7(a) is a schematic plan view of an exemplary mirror device, andFIG. 7(b) is a diagram illustrating a vibration model of the minordevice of FIG. 7(a).

FIGS. 8(a) and 8(b) are graphs showing relationships between numericalvalues in the case of outer drive.

FIGS. 9(a) and 9(b) are graphs showing a relationship between thenumerical values in the case of outer drive.

FIGS. 10(a) and 10(b) are graphs showing relationships between numericalvalues in the case of inner drive.

FIG. 11 is a plan view of a mirror device according to a firstmodification example.

FIG. 12 is a plan view of a minor device according to a secondmodification example.

DESCRIPTION OF EMBODIMENTS

Hereinafter, one embodiment of the present disclosure will be describedin detail with reference to the drawings. Incidentally, in the followingdescription, the same reference signs are used for the same orequivalent components and duplicated descriptions will be omitted.

First Embodiment

As illustrated in FIG. 1, a mirror device 1A of a first embodimentincludes a support portion 2, a first movable portion 3, a secondmovable portion 4, a pair of first connection portions 5 and 6, a pairof second connection portions 7 and 8, and a magnetic field generationunit 9. The support portion 2, the first movable portion 3, the secondmovable portion 4, the first connection portions 5 and 6, and the secondconnection portions 7 and 8 are integrally configured by, for example, asemiconductor substrate such as a silicon on insulator (SOI) substrate.Namely, the mirror device 1A is configured as a MEMS device.

In the mirror device 1A, the first movable portion 3 having a mirrorsurface (optical surface) 10 swings around each of an X-axis (firstaxis) and a Y-axis (second axis perpendicular to the first axis)orthogonal to each other. The mirror device 1A can be used, for example,in an optical switch for optical communication, an optical scanner, andthe like. The magnetic field generation unit 9 is configured by, forexample, permanent magnets in a Halbach array and the like.

The support portion 2 has, for example, a quadrilateral external shapein plan view, and is formed in a frame shape. The support portion 2 isdisposed on one side in a direction perpendicular to the X-axis and theY-axis with respect to the magnetic field generation unit 9. The firstmovable portion 3 is disposed inside the support portion 2 in a statewhere the first movable portion 3 is separated from the magnetic fieldgeneration unit 9. Incidentally, the “plan view” means when seen in thedirection perpendicular to the X-axis and the Y-axis, in other words,means when seen in a direction perpendicular to a plane on which thesupport portion 2, the first movable portion 3, the second movableportion 4, and the like are disposed.

The first movable portion 3 includes a main body portion 3 a and a frame3 b surrounding the main body portion 3 a. The main body portion 3 a isformed, for example, in an elliptical shape in plan view. For example,the mirror surface 10 having a circular shape is provided on a surfaceon an opposite side of the main body portion 3 a from the magnetic fieldgeneration unit 9. The mirror surface 10 is configured by, for example,a reflection film made of aluminum, an aluminum alloy, silver, a silveralloy, gold, a dielectric multilayer film, or the like. The frame 3 bhas, for example, a quadrilateral external shape in plan view, and isformed in a frame shape. The main body portion 3 a is connected to theframe 3 b on both sides on the Y-axis.

The second movable portion 4 has, for example, an octagonal externalshape in plan view, and is formed in a frame shape. The second movableportion 4 is disposed inside the support portion 2 to surround the firstmovable portion 3 in a state where the second movable portion 4 isseparated from the magnetic field generation unit 9.

The first connection portions 5 and 6 are disposed on both sides of thefirst movable portion 3 on the X-axis. Each of the first connectionportions 5 and 6 connects the first movable portion 3 and the secondmovable portion 4 to each other on the X-axis such that the firstmovable portion 3 is swingable around the X-axis (X-axis serving as acenter axis). Each of the first connection portions 5 and 6 is twistedaround the X-axis when the first movable portion 3 swings around theX-axis. Each of the first connection portions 5 and 6 extends linearly,for example, along the X-axis. Each of the first connection portions 5and 6 is formed, for example, in a plate shape that extends along theplane perpendicular to the X-axis and the Y-axis.

The first connection portion 5 includes a linear portion 5 a thatextends along the X-axis and has a constant width, and a pair of widenedportions 5 b connected to both ends of the linear portion 5 a. In planview, one widened portion 5 b is increased in width as approaching thefirst movable portion 3, and the other widened portion 5 b is increasedin width as approaching the second movable portion 4. A pair of edges ofeach of the widened portions 5 b are curved to protrude toward eachother.

The first connection portion 6 includes a linear portion 6 a thatextends along the X-axis and has a constant width, and a pair of widenedportions 6 b connected to both ends of the linear portion 6 a. In planview, one widened portion 6 b is increased in width as approaching thefirst movable portion 3, and the other widened portion 6 b is increasedin width as approaching the second movable portion 4. A pair of edges ofeach of the widened portions 6 b are curved to protrude toward eachother.

The second connection portions 7 and 8 are disposed on both sides of thesecond movable portion 4 on the Y-axis. Each of the second connectionportions 7 and 8 connects the second movable portion 4 and the supportportion 2 to each other on the Y-axis such that the second movableportion 4 is swingable around the Y-axis (Y-axis serving as a centeraxis). Each of the second connection portions 7 and 8 is twisted aroundthe Y-axis when the second movable portion 4 swings around the Y-axis.Each of the second connection portions 7 and 8 extends in a meanderingmanner in plan view. Each of the second connection portions 7 and 8includes a plurality of linear portions 11 a and a plurality of foldedportions 11 b. The plurality of linear portions 11 a extend along aY-axis direction, and are disposed side by side in an X-axis direction.The plurality of folded portions 11 b alternately connect both ends ofthe linear portions 11 a adjacent to each other.

The torsional spring constant of each of the second connection portions7 and 8 around the X-axis is smaller than the torsional spring constantof each of the first connection portions 5 and 6 around the X-axis. Thetorsional spring constant of each of the first connection portions 5 and6 around the X-axis is a ratio of the torsional moment around theX-axis, which is applied to each of the first connection portions 5 and6, to the torsional angle of each of the first connection portions 5 and6 around the X-axis. The torsional spring constant of each of the secondconnection portions 7 and 8 around the X-axis is a ratio of thetorsional moment around the X-axis, which is applied to each of thesecond connection portions 7 and 8, to the torsional angle of each ofthe second connection portions 7 and 8 around the X-axis.

The total length of each of the second connection portions 7 and 8 islonger than the total length of each of the first connection portions 5and 6. The total length of each of the first connection portions 5 and 6is the total length along an extending direction (direction along theX-axis in this example) of each of the first connection portions 5 and6. The total length of the second connection portion 7 is the totallength along an extending direction of the second connection portion 7,and in this example, is the total length obtained by adding the lengthsof portions including the linear portions 11 a and the folded portions11 b. The same point also applies to the total length of the secondconnection portion 8.

The mirror device 1A further includes coils (drive coils) 21 and 22, aplurality of wirings 12, 13, 14, and 15, and a plurality of electrodepads 31, 32, 33, and 34. The coil 21 is embedded, for example, in theframe 3 b of the first movable portion 3, and extends spirally in planview. The coil 22 is embedded, for example, in the second movableportion 4, and extends spirally in plan view. The coils 21 and 22 aremade of a metallic material such as copper. In FIG. 1, the regions ofdisposition of the coils 21 and 22 are illustrated by hatching.

The plurality of electrode pads 31, 32, 33, and 34 are provided on thesupport portion 2, and are exposed to the outside from an insulationlayer 23. The insulation layer 23 is integrally formed to cover thesurfaces (surfaces opposite to the magnetic field generation unit 9) ofthe support portion 2, the first movable portion 3, the second movableportion 4, the first connection portions 5 and 6, and the secondconnection portions 7 and 8. The insulation layer 23 is made of, forexample, silicon dioxide, silicon nitride, or the like.

The wiring 12 electrically connects one end of the coil 21 and theelectrode pad 31. The wiring 12 extends from the one end of the coil 21to the electrode pad 31 via the first connection portion 5, the secondmovable portion 4, and the second connection portion 7. The wiring 13electrically connects the other end of the coil 21 and the electrode pad32. The wiring 13 extends from the other end of the coil 21 to theelectrode pad 32 via the first connection portion 6, the second movableportion 4, and the second connection portion 8. Portions 12 a and 13 aof the wirings 12 and 13 are made of a metallic material such astungsten, the portions 12 a and 13 a passing through the firstconnection portions 5 and 6. The other portions of the wirings 12 and 13are made of a metallic material such as aluminum, an aluminum alloy, orcopper. In FIG. 1, the portions 12 a and 13 a are illustrated byhatching.

The wiring 14 electrically connects one end of the coil 22 and theelectrode pad 33. The wiring 14 extends from the one end of the coil 22to the electrode pad 33 via the second connection portion 7. The wiring15 electrically connects the other end of the coil 22 and the electrodepad 34. The wiring 15 extends from the other end of the coil 22 to theelectrode pad 34 via the second connection portion 8.

In the mirror device 1A configured as described above, when a drivesignal for linear operation is input to the coil 22 via the electrodepads 33 and 34 and the wirings 14 and 15, a Lorentz force is applied tothe coil 22 by interaction with a magnetic field generated by themagnetic field generation unit 9. The balance between the Lorentz forceand the elastic forces of the second connection portions 7 and 8 can beused to cause the mirror surface 10 (first movable portion 3) tolinearly operate around the Y-axis, together with the second movableportion 4.

Meanwhile, when a drive signal for resonance operation is input to thecoil 21 via the electrode pads 31 and 32 and the wirings 12 and 13, aLorentz force is applied to the coil 21 by interaction with a magneticfield generated by the magnetic field generation unit 9. In addition tothe Lorentz force, the resonance of the first movable portion 3 at theresonance frequency can be used to cause the mirror surface 10 (firstmovable portion 3) to resonate around the X-axis. Namely, in the mirrordevice 1A, the coil 21 and the magnetic field generation unit 9 form adrive unit that directly applies a drive force (Lorentz force) to thefirst movable portion 3, and the coil 22 and the magnetic fieldgeneration unit 9 form a drive unit that directly drives a drive force(Lorentz force) to the second movable portion 4.

Second Embodiment

In the mirror device 1A of the first embodiment, the first movableportion 3 swings around each of the X-axis and the Y-axis whereas in amirror device 1B of a second embodiment illustrated in FIG. 2, the firstmovable portion 3 swings only around the X-axis. In the mirror device1B, the main body portion 3 a of the first movable portion 3 is formedin a circular shape in plan view. The main body portion 3 a is connectedto the frame 3 b on both sides on the X-axis and on both sides on theY-axis. The second movable portion 4 has a quadrilateral external shapein plan view.

The second connection portions 7 and 8 are disposed on both sides of thesecond movable portion 4 on the X-axis. The second connection portions 7and 8 connect the second movable portion 4 and the support portion 2 toeach other on the X-axis such that the second movable portion 4 isvibrated to cause the first movable portion 3 to be swingable around theX-axis. Each of the second connection portions 7 and 8 is twisted aroundthe X-axis when the second movable portion 4 vibrates.

The mirror device 1B does not include the coil 21, the wirings 12 and13, and the electrode pads 31 and 32, and includes the coil 22, a coil(electromotive force monitoring coil or sensing coil) 24, a plurality ofwirings 14, 15, 16, and 17, and a plurality of electrode pads 33, 34,35, and 36. The coil 24 is embedded, for example, in the frame 3 b ofthe first movable portion 3, and extends spirally in plan view. The coil24 is made of a metallic material such as copper.

The wiring 16 electrically connects one end of the coil 24 and theelectrode pad 35. The wiring 17 extends from the one end of the coil 24to the electrode pad 35 via the first connection portion 5, the secondmovable portion 4, and the second connection portion 7. The wiring 17electrically connects the other end of the coil 24 and the electrode pad36. The wiring 17 extends from the other end of the coil 24 to theelectrode pad 36 via the first connection portion 6, the second movableportion 4, and the second connection portion 8.

In the mirror device 1B, when a drive signal for resonance operation isinput to the coil 22 via the electrode pads 33 and 34 and the wirings 14and 15, a Lorentz force is applied to the coil 22 by interaction with amagnetic field generated by the magnetic field generation unit 9. Inaddition to the Lorentz force, the resonance of the first movableportion 3 at the resonance frequency can be used to cause the mirrorsurface 10 (first movable portion 3) to resonate around the X-axis.Specifically, when a drive signal of a frequency equal to the resonancefrequency of the first movable portion 3 around the X-axis is input tothe coil 22, the second movable portion 4 vibrates slightly around theX-axis at the frequency. The vibration is transmitted to the firstmovable portion 3 via the first connection portions 5 and 6, so that thefirst movable portion 3 can swing around the X-axis at the frequency.

In addition, in the mirror device 1B, during operation of the firstmovable portion 3, an electromotive force is generated in the coil 24 byinteraction with a magnetic field generated by the magnetic fieldgeneration unit 9. For this reason, a signal output from the electrodepads 35 and 36 via the wirings 16 and 17 is monitored, so that thedeflection angle of the first movable portion 3, namely, the deflectionangle of the mirror surface 10 can be identified. In such a manner, thecoil 24 serves as an electromotive force monitoring coil.

Subsequently, the configuration of each part of the mirror device 1Bwill be further described with reference to FIGS. 2 and 3. Hereinafter,a structure around the first connection portion 5 and the secondconnection portion 7 will be described, and the first connection portion6 and the second connection portion 8 are also similarly configured.

The second connection portion 7 includes a first member 7 a and a pairof second members 7 b. The first member 7 a and the pair of secondmembers 7 b extend along the X-axis to connect the second movableportion 4 and the support portion 2 to each other. For example, thefirst member 7 a and the pair of second members 7 b are disposed side byside along a direction parallel to the Y-axis. The first member 7 a isdisposed on the X-axis, and the pair of second members 7 b are disposedto interpose the first member 7 a therebetween. For example, the firstmember 7 a and the pair of second members 7 b are disposed side by sideat equal intervals to be close to each other. The first member 7 a andthe pair of second members 7 b are formed, for example, in the sameshape. Each of the second members 7 b serves as a vibration suppressingportion that suppresses the vibration of the second movable portion 4around the X-axis.

The support portion 2 is provided with a recessed portion 2 a located onthe X-axis, and the first member 7 a and the pair of second members 7 bare connected to the support portion 2 in the recessed portion 2 a. As aresult, while the lengths of the first member 7 a and the pair of secondmembers 7 b are secured, the size of the mirror device 1B can bereduced.

The torsional spring constant of the second connection portion 7 aroundthe X-axis is smaller than the torsional spring constant of the firstconnection portion 5 around the X-axis. The torsional spring constant ofthe second connection portion 7 around the X-axis is the torsionalspring constant of the entirety of the second connection portion 7,which includes the first member 7 a and the pair of second members 7 b,around the X-axis. The total length of the second connection portion 7is shorter than the total length of the first connection portion 5. Forexample, the total length of the second connection portion 7 is shorterthan ½ of the total length of the first connection portion 5. In thisexample, the total length of the second connection portion 7 is thetotal length obtained by adding the lengths of the first member 7 a andthe pair of second members 7 b. In this example, the first member 7 aand the pair of second members 7 b are equal to each other in length.

A portion of the wiring 16, the portion being located on the firstconnection portion 5, is configured by a diffusion region 41 whereimpurities are diffused in a semiconductor material. The diffusionregion 41 is formed by diffusing n-type impurities on the surface of ap-type silicon substrate. The wiring 16 further includes an innerportion 16 a that electrically connects one end portion 41 a of thediffusion region 41 and the coil 24, and an outer portion 16 b thatelectrically connects the other end portion 41 b of the diffusion region41 and the coil 24.

The diffusion region 41 extends from the first connection portion 5 tothe first movable portion 3, and is connected to the inner portion 16 aon the first movable portion 3. In addition, the diffusion region 41extends from the first connection portion 5 to the second movableportion 4, and is connected to the outer portion 16 b on the secondmovable portion 4. The outer portion 16 b extends to pass through thefirst member 7 a of the second connection portion 7 to be connected tothe electrode pad 35. Similarly, the wiring 14 extending from the coil22 also extends to pass through the first member 7 a of the secondconnection portion 7 to be connected to the electrode pad 33.

The end portion 41 a is increased in width as approaching the coil 24,and the end portion 41 b is increased in width as approaching the coil22. The end portions 41 a and 41 b have external shapes along a pair ofwidened portions 5 b of the first connection portion 5. A width W1 ofthe end portion 41 a and a width W2 of the end portion 41 b are widerthan a width W3 of the first connection portion 5. Namely, each of thewidths W1 and W2 is wider than the width of the diffusion region 41 onthe first connection portion 5. The width of each part of the diffusionregion 41 is the maximum width in plan view, and is a width in adirection perpendicular to an extending direction of the diffusionregion 41 (direction parallel to the Y-axis in this example). The widthof the first connection portion 5 is the minimum width in plan view, andfor example, is the width of the linear portion 5 a in a directionperpendicular to an extending direction of the first connection portion5 (direction parallel to the Y-axis in this example). Incidentally, thefirst connection portion 5 is a portion that is twisted around theX-axis when the first movable portion 3 swings around the X-axis, and inthe mirror device 1B, the first connection portion 5 includes the pairof widened portions 5 b.

As illustrated in FIG. 4, the first movable portion 3 is provided with agroove 42 a having a shape corresponding to the coil 24, and the secondmovable portion 4 is provided with a groove 42 b having a shapecorresponding to the coil 22. Insulation layers 43 and 44 are providedon inner surfaces of the grooves 42 a and 42 b, and insulation layers 45and 46 are provided on the insulation layer 44. The coil 24 is disposedin the groove 42 with the insulation layers 43 and 44 interposedtherebetween.

Each of the insulation layers 43 to 46 is made of, for example, silicondioxide, silicon nitride, or the like. The inner portion 16 a and theouter portion 16 b of the wiring 16 are disposed on the insulation layer45. The inner portion 16 a and the outer portion 16 b are formed toenter an opening 47 penetrating through the insulation layers 43 to 45,and are connected to the diffusion region 41 in the opening 47.

Each of the inner portion 16 a and the outer portion 16 b includes afirst layer 48 and a second layer 49 disposed on the first layer 48. Thefirst layer 48 serves as a contact layer in contact with the diffusionregion 41, and also serves as a barrier layer that prevents thediffusion of a metal element into the silicon substrate, the metalelement being contained in the second layer 49. The first layer 48 ismade of a metallic material such as tungsten, titanium nitride,molybdenum, tantalum, or tantalum nitride. The second layer 49 is madeof, for example, an aluminum alloy or copper.

A width W4 of a contact portion between the diffusion region 41 and theinner portion 16 a and a width W5 of a contact portion between thediffusion region 41 and the outer portion 16 b are wider than the widthW3 of the first connection portion 5. The widths of the contact portionsare widths in the direction perpendicular to the extending direction ofthe first connection portion 5 (direction parallel to the Y-axis in thisexample) in plan view.

[Method for Producing Mirror Device]

Subsequently, a method (design method) for producing the minor devices1A and 1B as described above will be described. FIG. 5(a) is a graphshowing an example of a relationship between the angular frequency ofthe drive signal and the deflection angles of the first movable portion3 and the second movable portion 4 around the X-axis, and FIG. 5(b) is agraph showing another example of the relationship. In FIGS. 5(a) and5(b), the deflection angle of the first movable portion 3 is illustratedby a solid line, and the deflection angle of the second movable portion4 is illustrated by a broken line.

As shown in FIGS. 5(a) and 5(b), in the mirror devices 1A and 1B, a twodegree of freedom system including the support portion 2, the firstmovable portion 3, the second movable portion 4, the first connectionportions 5 and 6, and the second connection portions 7 and 8 has tworesonance points A and B for vibration around the X-axis. It should benoted that in the two degree of freedom system, only the vibrationaround the X-axis is taken into consideration, and vibration around theY-axis is not taken into consideration. A natural angular frequency ω₁at the resonance point A and a natural angular frequency (02 at theresonance point B (where ω₁<ω₂) are natural angular frequencies for thevibration of the first movable portion 3 and the second movable portion4 around the X-axis. The resonance point A can be used to operate themirror surface 10 at a relatively small frequency. The resonance point Bcan be used to operate the mirror surface 10 at a relatively largefrequency.

The relationship between the deflection angles of the first movableportion 3 and the second movable portion 4 when the mirror surface 10 isoperated at the resonance point A of FIG. 5(a) can be illustrated as inFIG. 6(a), and the relationship between the deflection angles of thefirst movable portion 3 and the second movable portion 4 when the mirrorsurface 10 is operated at the resonance point B of FIG. 5(a) can beillustrated as in FIG. 6(b).

As illustrated in FIG. 6(a), when the minor surface 10 is operated atthe resonance point A of FIG. 5(a), the first movable portion 3 and thesecond movable portion 4 rotate to the same extent, so that thedifference between the deflection angles of the first movable portion 3and the second movable portion 4 is small. Namely, the first connectionportions 5 and 6 are hardly twisted. As illustrated in FIG. 6(b), whenthe mirror surface 10 is operated at the resonance point B of FIG. 5(a),whereas the second movable portion 4 hardly rotates, the first movableportion 3 rotates, so that the difference between the deflection anglesof the first movable portion 3 and the second movable portion 4 islarge. Namely, the first connection portions 5 and 6 are efficientlytwisted.

Contrary to the example of FIG. 5(a), in the example of FIG. 5(b), whenthe mirror surface 10 is operated at the resonance point A, whereas thesecond movable portion 4 hardly rotates, the first movable portion 3rotates, and when the minor surface 10 is operated at the resonancepoint B, the first movable portion 3 and the second movable portion 4rotate to the same extent. From the viewpoint of high efficiency, whenthe mirror surface 10 is operated at either of the resonance points Aand B, it is preferable that whereas the deflection angle of the firstmovable portion 3 is large, the deflection angle of the second movableportion 4 is small Namely, it is preferable that a ratio of thedeflection angle of the first movable portion 3 to the deflection angleof the second movable portion 4 (hereinafter, referred to as an“amplitude ratio”) is greatly secured. Hereinafter, a method forproducing the minor devices 1A and 1B, which greatly secures theamplitude ratio, will be described.

[Case of Outer Drive]

First, as in the mirror device 1B, a case where a drive force is appliedto the second movable portion 4 to cause the first movable portion 3 toresonate (case of outer drive) will be described. FIG. 7(a) is aschematic plan view of a mirror device 1C that is an exemplary example,and FIG. 7(b) is a diagram illustrating a vibration model of the mirrordevice 1C. The vibration of the first movable portion 3 and the secondmovable portion 4 around the X-axis can be modeled by the vibrationmodel illustrated in FIG. 7(b). The equations of motion for thevibration of the first movable portion 3 and the second movable portion4 around the X-axis are expressed by the following equations (9) and(10).

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 9} \right\rbrack & \; \\{{j_{i}\frac{d^{2}\theta_{i}}{{dt}^{2}}} + {c_{i}\frac{d\;\theta_{i}}{dt}} + {k_{i}\left( {{\theta_{i} - \left. \theta_{0} \right)} = {{T_{i}\cos\;\left( {\omega\; t} \right)} = 0}} \right.}} & (9) \\\left\lbrack {{Equation}\mspace{14mu} 10} \right\rbrack & \; \\{{{j_{o}\frac{d^{2}\theta_{o}}{dt^{2}}} + {c_{o}\frac{d\;\theta_{o}}{dt}} + {\left( {k_{i} - k_{0}} \right)\theta_{0}} - {k_{i}\theta_{i}}} = {T_{o}{\cos\left( {\omega\; t} \right)}}} & (10)\end{matrix}$

In the above equations, θ_(i) is the deflection angle (rad) of the firstmovable portion 3 (deflection angle of each of the first connectionportions 5 and 6), j_(i) is the inertia moment (kgm²) of the firstmovable portion 3, k_(i) is the torsional spring constant (Nm/rad) ofthe first connection portions 5 and 6, c_(i) is the viscous dampingcoefficient (Ns/m) of the first movable portion 3, T_(i) is torque (Nm)applied to the first movable portion 3, θ_(o) is the deflection angle(rad) of the second movable portion 4 (deflection angle of each of thesecond connection portions 7 and 8), j_(o) is the inertia moment (kgm²)of the second movable portion 4, k_(o) is the torsional spring constant(Nm/rad) of each of the second connection portions 7 and 8, c_(o) is theviscous damping coefficient (Ns/m) of the second movable portion 4,T_(o) is torque (Nm) applied to the second movable portion 4, ω is theangular frequency (rad/s) of the drive signal, and t is time (s). Thevalues are values for vibration around the X-axis. The viscous dampingcoefficients c_(i) and c_(o) are expressed by the following equations(11) and (12). Here, is the damping ratio of the first movable portion3, and is the damping ratio of the second movable portion 4.

[Equation 11]

c _(i)=2ζ_(i)√{square root over (j _(i) k _(i))}  (11)

[Equation 12]

c _(o)=2ζ_(o)√{square root over (j _(o) k _(o))}  (11)

In the case of outer drive, since a drive force is not applied to thefirst movable portion 3 (namely, since T_(i)=0), the right side ofequation (9) is 0.

The deflection angle θ_(i) of the first movable portion 3 and thedeflection angle θ_(o) of the second movable portion 4 are obtained fromequations (9) and (10) as expressed in the following equations (13) and(14).

$\begin{matrix}{\mspace{79mu}\left\lbrack {{Equation}\mspace{14mu} 13} \right\rbrack} & \; \\{\mspace{79mu}{\theta_{i} = {\frac{T_{o}}{j_{o}\omega_{oo}^{2}}\sqrt{\frac{B^{2} + C^{2}}{\left( {{\beta_{io}^{2}B} - \beta_{io}^{2} - 1 + \beta_{oo}^{2}} \right)^{2} + \left( {{\beta_{io}^{2}C} + {2\beta_{oo}\zeta_{o}}} \right)^{2}}}}}} & (13) \\{\mspace{79mu}\left\lbrack {{Equation}\mspace{14mu} 14} \right\rbrack} & \; \\{{{\theta_{o} = {\frac{T_{o}}{j_{o}\omega_{oo}^{2}}\sqrt{\frac{1}{\left( {{\beta_{io}^{2}B} - \beta_{io}^{2} - 1 + \beta_{oo}^{2}} \right)^{2} + \left( {{\beta_{io}^{2}C} + {2\beta_{oo}\zeta_{o}}} \right)^{2}}}}}\mspace{20mu}{where}}\mspace{79mu}} & (14) \\{\mspace{79mu}\left\lbrack {{Equation}\mspace{14mu} 15} \right\rbrack} & \; \\{\mspace{79mu}{B = \frac{1 - \beta_{ii}^{2}}{\left( {1 - B_{ii}^{2}} \right)^{2} + {4\beta_{ii}^{2}\zeta_{i}^{2}}}}} & (15) \\{\mspace{79mu}\left\lbrack {{Equation}\mspace{14mu} 16} \right\rbrack} & \; \\{\mspace{79mu}{C = \frac{4\beta_{ii}^{2}\zeta_{i}^{2}}{\left( {1 - \beta_{ii}^{2}} \right)^{2} + {4\beta_{ii}^{2}\zeta_{i}^{2}}}}} & (16) \\{\mspace{79mu}\left\lbrack {{Equation}\mspace{14mu} 17} \right\rbrack} & \; \\{\mspace{79mu}{\beta_{ii} = {\frac{\omega}{\omega_{ii}} = \frac{\omega}{\sqrt{\frac{k_{i}}{j_{i}}}}}}} & (17) \\{\mspace{79mu}\left\lbrack {{Equation}\mspace{14mu} 18} \right\rbrack} & \; \\{\mspace{79mu}{\beta_{oo} = {\frac{\omega}{\omega_{oo}} = \frac{\omega}{\sqrt{\frac{k_{o}}{j_{o}}}}}}} & (18) \\{\mspace{79mu}\left\lbrack {{Equation}\mspace{14mu} 19} \right\rbrack} & \; \\{\mspace{79mu}{\beta_{io} = {\frac{\omega_{io}}{\omega_{oo}} = {\frac{\sqrt{\frac{k_{i}}{j_{o}}}}{\sqrt{\frac{k_{o}}{j_{o}}}} = \sqrt{\frac{k_{i}}{k_{o}}}}}}} & (19)\end{matrix}$

In this case, the amplitude ratio θ_(i)/θ_(o) is expressed by thefollowing equation (20). Incidentally, ζ_(o) is, for example, 0.01 orless, which is extremely smaller than 1.

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 20} \right\rbrack & \; \\{\frac{\theta_{i}}{\theta_{o}} = {\sqrt{\frac{1}{\left( {1 - \beta_{ii}^{2}} \right)^{2} + {4\beta_{ii}^{2}\zeta_{i}^{2}}}} \approx {\sqrt{\frac{1}{\left\{ {1 - \left( \frac{\omega}{\omega_{ii}} \right)^{2}} \right\}^{2}}}\left( {\because{\zeta_{i} ⪡ 1}} \right)}}} & (20)\end{matrix}$

Therefore, the condition for setting the amplitude ratio to 5 or more isexpressed by the following equation (21).

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 21} \right\rbrack & \; \\{\frac{\theta_{i}}{\theta_{o}} \approx \sqrt{\frac{1}{\left\{ {1 - \left( \frac{\omega}{\omega_{ii}} \right)^{2}} \right\}^{2}}} \geq 5} & (21)\end{matrix}$

When the mirror surface 10 is operated at the natural angular frequencyω₁ of the resonance point A (namely, when ω=ω₁), since 1−(ω₁/ω_(ii))² islarger than 0, the condition for setting the amplitude ratio to 5 ormore is expressed by the following equation (22).

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 22} \right\rbrack & \; \\{0 < {1 - \left( \frac{\omega_{1}}{\omega_{ii}} \right)^{2}} \leq {0.2}} & (22)\end{matrix}$

When the mirror surface 10 is operated at the natural angular frequencyω₂ of the resonance point B (namely, when ω=ω₂), since 1−(ω₂/ω_(ii))² issmaller than 0, the condition for setting the amplitude ratio to 5 ormore is expressed by the following equation (23).

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 23} \right\rbrack & \; \\{0 < {\left( \frac{\omega_{2}}{\omega_{ii}} \right)^{2} - 1} \leq 0.2} & (23)\end{matrix}$

Namely, when the mirror surface 10 is operated at the natural angularfrequency ω₁, the mirror device is designed and produced to satisfyequation (22), so that the amplitude ratio can be 5 or more, and whenthe mirror surface 10 is operated at the natural angular frequency ω₂,the mirror device is designed and produced to satisfy equation (23), sothat the amplitude ratio can be 5 or more.

FIGS. 8(a) and 8(b) are graphs showing relationships between numericalvalues when the mirror surface 10 is operated at the natural angularfrequency ω₁ in the case of outer drive. FIG. 8(a) shows a relationshipbetween 1−(ω₁/ω_(ii))² and the amplitude ratio θ_(i)/θ_(o), and FIG.8(b) shows a relationship between the torsional spring constant k_(o) ofeach of the second connection portions 7 and 8 and the amplitude ratioθ_(i)/θ_(o). As shown in FIG. 8(a), when the minor device is designedand produced to satisfy equation (22), the amplitude ratio can be 5 ormore. As shown in FIG. 8(b), as the torsional spring constant k_(o)increases, the amplitude ratio increases. Therefore, the torsionalspring constant k_(o) is increased, so that the amplitude ratio can begreatly secured.

FIGS. 9(a) and 9(b) are graphs showing relationships between numericalvalues when the minor surface 10 is operated at the natural angularfrequency ω₂ in the case of outer drive. FIG. 9(a) shows a relationshipbetween 1−(ω₂/ω_(ii))² and the amplitude ratio θ_(i)/θ_(o), and FIG.9(b) shows a relationship between the torsional spring constant k_(o) ofthe second connection portions 7 and 8 and the amplitude ratioθ_(i)/θ_(o). As shown in FIG. 9(a), when the minor device is designedand produced to satisfy equation (23), the amplitude ratio can be 5 ormore. As shown in FIG. 9(b), as the torsional spring constant k_(o)increases, the amplitude ratio decreases. Therefore, the torsionalspring constant k_(o) is decreased, so that the amplitude ratio can begreatly secured.

Here, when the values of the natural angular frequencies ω₁ and ω₂ areclose to each other, and the difference between the natural angularfrequencies ω₁ and ω₂ is small, both of equations (22) and (23) aresatisfied. However, when the difference between the natural angularfrequencies ω₁ and ω₂ is small, for example, when the minor surface 10is operated at the natural angular frequency ω₁, a resonance response isgenerated at the natural angular frequency ω₂, so that the operation ofthe mirror surface 10 is unstable. Therefore, in the method forproducing the mirror device of the present embodiment, in the case ofouter drive, the mirror device is designed and produced to satisfy onlyone of equations (22) and (23) but not to satisfy the other. As aresult, the difference between the natural angular frequencies ω₁ and ω₂can be greatly secured, and an unnecessary resonance response can besuppressed to stabilize the operation. This point is common to when themirror surface 10 is operated at either of the natural angularfrequencies ω₁ and ω₂.

For example, in the mirror device 1B according to the second embodiment,only equation (22) is satisfied and equation (23) is not satisfied.Therefore, the mirror surface 10 is operated at the natural angularfrequency ω₁, so that the amplitude ratio can be 5 or more. In addition,the difference between the natural angular frequencies ω₁ and ω₂ can begreatly secured, and an unnecessary resonance response can be suppressedto stabilize the operation.

[Case of Inner Drive]

Next, as in the mirror device 1A, a case where a drive force is appliedto the first movable portion 3 to cause the first movable portion 3 toresonate (case of inner drive) will be described. In this case, theequations of motion for the vibration of the first movable portion 3 andthe second movable portion 4 around the X-axis are expressed by thefollowing equations (24) and (25).

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 24} \right\rbrack & \; \\{{{j_{i}\frac{d^{2}\theta_{i}}{{dt}^{2}}} + {c_{i}\frac{d\theta_{i}}{dt}} + {k_{i}\left( {\theta_{i} - \theta_{0}} \right)}} = {T_{i}{\cos\left( {\omega t} \right)}}} & (24) \\\left\lbrack {{Equation}\mspace{14mu} 25} \right\rbrack & \; \\{{{j_{o}\frac{d^{2}\theta_{o}}{{dt}^{2}}} + {c_{o}\frac{d\;\theta_{o}}{dt}} + {\left( {k_{i} - k_{0}} \right)\theta_{0}} - {k_{i}\theta_{i}}} = {{T_{o}{\cos\left( {\omega\; t} \right)}} = 0}} & (25)\end{matrix}$

In the case of inner drive, since a drive force for driving the firstmovable portion 3 is not applied to the second movable portion 4(namely, since T_(o)=0), the right side of equation (25) is 0.

The deflection angle θ_(i) of the first movable portion 3 and thedeflection angle θ_(o) of the second movable portion 4 are obtained fromequations (24) and (25) as expressed in the following equations (26) and(27).

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 26} \right\rbrack & \; \\{\theta_{i} = {\frac{T_{i}}{j_{i}\omega_{ii}^{2}}\sqrt{\frac{1}{\left( {{- \beta_{ii}^{2}} + 1 + B} \right)^{2} + \left( {{2\beta_{ii}\zeta_{i}} + C} \right)^{2}}}}} & (26) \\\left\lbrack {{Equation}\mspace{14mu} 27} \right\rbrack & \; \\{\theta_{o} = {\frac{T_{i}}{j_{i}\omega_{ii}^{2}}\sqrt{\frac{B^{2} + C^{2}}{\left( {{- \beta_{ii}^{2}} + 1 + B} \right)^{2} + \left( {{2\beta_{ii}\zeta_{i}} + C} \right)^{2}}}}} & (27) \\{{where}\left\lbrack {{Equation}\mspace{14mu} 28} \right\rbrack} & \; \\{B = \frac{\beta_{io}^{2}\left( {1 - \beta_{oo}^{2} + \beta_{io}^{2}} \right)}{\left( {1 - \beta_{00}^{2} + \beta_{io}^{2}} \right)^{2} + {4\beta_{oo}^{2}\zeta_{o}^{2}}}} & (28) \\\left\lbrack {{Equation}\mspace{14mu} 29} \right\rbrack & \; \\{C = \frac{2\beta_{oo}\beta_{io}^{2}}{\left( {1 - \beta_{oo}^{2} + \beta_{io}^{2}} \right)^{2} + {4\beta_{oo}^{2}\zeta_{o}^{2}}}} & (29)\end{matrix}$

In this case, the amplitude ratio θ_(i)/θ_(o) is expressed by thefollowing equation (30).

$\begin{matrix}{\mspace{79mu}\left\lbrack {{Equation}\mspace{14mu} 30} \right\rbrack} & \; \\{{\frac{\theta_{i}}{\theta_{o}} \approx \frac{1}{\sqrt{\frac{\beta_{io}^{4}}{\left( {1 - \beta_{oo}^{2} + \beta_{io}^{2}} \right)^{2} + {4\beta_{oo}^{2}\zeta_{o}^{2}}}}} \approx \frac{1}{\sqrt{\frac{\beta_{io}^{4}}{\left( {1 - \beta_{oo}^{2} + \beta_{io}^{2}} \right)^{2}}}}} = {\frac{1}{\sqrt{\frac{\left( \frac{\omega_{io}}{\omega_{oo}} \right)^{4}}{\left\{ {1 - \left( \frac{\omega_{io}}{\omega_{oo}} \right)^{2} + \left( \frac{\omega}{\omega_{oo}} \right)^{2}} \right\}^{2}}}}\left( {\because{\zeta_{o} ⪡ 1}} \right)}} & (30)\end{matrix}$

Therefore, the condition for setting the amplitude ratio to 5 or more isexpressed by the following equation (31).

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 31} \right\rbrack & \; \\{\frac{\theta_{i}}{\theta_{o}} \approx \frac{1}{\sqrt{\frac{\left( \frac{\omega_{io}}{\omega_{oo}} \right)^{4}}{\left\{ {1 - \left( \frac{\omega_{io}}{\omega_{oo}} \right)^{2} + \left( \frac{\omega}{\omega_{oo}} \right)^{2}} \right\}^{2}}}} \geq 5} & (31)\end{matrix}$

When the mirror surface 10 is operated at the natural angular frequencyω₁ of the resonance point A (namely, when ω=ω₁), since1−(ω_(io)/ω_(oo))²+(ω₁/ω_(oo))² is larger than 0, the condition forsetting the amplitude ratio to 5 or more is expressed by the followingequation (32).

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 32} \right\rbrack & \; \\{0 < \frac{4\omega_{io}^{2}}{\omega_{oo}^{2} - \omega_{1}^{2}} \leq 1} & (32)\end{matrix}$

Meanwhile, when the mirror surface 10 is operated at the natural angularfrequency ω₂ of the resonance point B (namely, when ω=ω₂), since1−(ω_(io)/ω_(oo))²+(ω₂/ω_(oo))² is smaller than 0, the condition forsetting the amplitude ratio to 5 or more is expressed by the followingequation (33).

$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 33} \right\rbrack & \; \\{\frac{\omega_{oo}^{2} - \omega_{2}^{2}}{\omega_{io}^{2}} \leq {- 6}} & (33)\end{matrix}$

Namely, when the mirror surface 10 is operated at the natural angularfrequency ω₁, the mirror device is designed and produced to satisfyequation (32), so that the amplitude ratio can be 5 or more, and whenthe mirror surface 10 is operated at the natural angular frequency ω₂,the mirror device is designed and produced to satisfy equation (33), sothat the amplitude ratio can be 5 or more. The point that the mirrordevice is designed and produced to satisfy only one of equations (32)and (33) but not to satisfy the other, so that the difference betweenthe natural angular frequencies ω₁ and ω₂ can be greatly secured, and anunnecessary resonance response can be suppressed to stabilize theoperation is the same as in the case of outer drive.

FIGS. 10(a) and 10(b) are graphs showing relationships between numericalvalues in the case of inner drive. FIG. 10(a) shows a relationshipbetween 4ω_(io) ²/(ω_(oo) ²−ω₁ ²) and the amplitude ratio θ_(i)/θ_(o)when the mirror surface 10 is operated at the natural angular frequencyω₁, and FIG. 10(b) shows a relationship between (ω_(oo) ²−ω₂ ²)/ω_(io) ²and the amplitude ratio θ_(i)/θ_(o) when the mirror surface 10 isoperated at the natural angular frequency ω₂.

As shown in FIG. 10(a), when the minor device is designed and producedto satisfy equation (32), the amplitude ratio can be 5 or more. Inaddition, in the case of inner drive, when the minor surface 10 isoperated at the natural angular frequency ω₁, similarly to the case ofouter drive (FIG. 8(b)), as the torsional spring constant k_(o)increases, the amplitude ratio increases. Therefore, the torsionalspring constant k_(o) is increased, so that the amplitude ratio can begreatly secured.

As shown in FIG. 10(b), when the minor device is designed and producedto satisfy equation (33), the amplitude ratio can be 5 or more. Inaddition, in the case of inner drive, when the minor surface 10 isoperated at the natural angular frequency ω₂, similarly to the case ofouter drive (FIG. 9(b)), as the torsional spring constant k_(o)increases, the amplitude ratio decreases. Therefore, the torsionalspring constant k_(o) is decreased, so that the amplitude ratio can begreatly secured.

For example, in the mirror device 1A according to the first embodiment,only equation (33) is satisfied and equation (32) is not satisfied.Therefore, the mirror surface 10 is operated at the natural angularfrequency ω₂, so that the amplitude ratio can be 5 or more. In addition,the difference between the natural angular frequencies ω₁ and ω₂ can begreatly secured, and an unnecessary resonance response can be suppressedto stabilize the operation.

[Functions and Effects]

As described above, in the minor device 1A, since two natural angularfrequencies ω₁ and ω₂ satisfy only equation (33) (first equation) and donot satisfy equation (32) (second equation), the first movable portion 3and the second movable portion 4 swing at the natural angular frequencyω₂, so that the amplitude ratio can be 5 or more. Therefore, theamplitude ratio can be greatly secured, and the efficiency of theoperation can be improved. Furthermore, since only equation (33) issatisfied and equation (32) is not satisfied, the difference between thenatural angular frequencies ω₁ and ω₂ can be greatly secured, and anunnecessary resonance response can be suppressed to stabilize theoperation. As a result, according to the mirror device 1A, a highlyefficient and stable operation can be implemented.

In the mirror device 1B, since the two natural angular frequencies ω₁and ω₂ satisfy only equation (22) (first equation) and do not satisfyequation (23) (second equation), the first movable portion 3 and thesecond movable portion 4 swing at the natural angular frequency ω₁, sothat the amplitude ratio can be 5 or more. Therefore, the amplituderatio can be greatly secured, and the efficiency of the operation can beimproved. Furthermore, since only equation (22) is satisfied andequation (23) is not satisfied, the difference between the naturalangular frequencies ω₁ and ω₂ can be greatly secured, and an unnecessaryresonance response can be suppressed to stabilize the operation. As aresult, according to the mirror device 1B, a highly efficient and stableoperation can be implemented.

In the mirror device 1A, each of the second connection portions 7 and 8connects the second movable portion 4 and the support portion 2 to eachother such that the second movable portion 4 is swingable around theY-axis. For this reason, the second movable portion 4 is swingablearound the Y-axis, together with the first movable portion 3.

In the mirror device 1B, the torsional spring constant of each of thesecond connection portions 7 and 8 around the X-axis is larger than thetorsional spring constant of each of the first connection portions 5 and6 around the X. For this reason, the amplitude ratio can be more greatlysecured.

In the mirror device 1B, the total length of each of the secondconnection portions 7 and 8 is shorter than the total length of each ofthe first connection portions 5 and 6. For this reason, the torsionalspring constant of the second connection portions 7 and 8 can besuitably made larger than the torsional spring constant of the firstconnection portions 5 and 6. Furthermore, instead of increasing thecross-sectional areas of the second connection portions 7 and 8, thelengths of the second connection portions 7 and 8 are shortened toincrease the torsional spring constant of the second connection portions7 and 8; and thereby, while the size of the mirror device 1B is reduced,the amplitude ratio can be greatly secured.

In the mirror device 1B, the first member 7 a and the pair of secondmembers 7 b (a plurality of members) are disposed side by side along thedirection parallel to the Y-axis. For this reason, the torsional springconstant of the second connection portions 7 and 8 can be more suitablymade larger than the torsional spring constant of the first connectionportions 5 and 6.

In the mirror device 1B, each of the second connection portions 7 and 8includes the first member 7 a disposed on the X-axis, and the pair ofsecond members 7 b disposed to interpose the first member 7 atherebetween. For this reason, the torsional spring constant of thesecond connection portions 7 and 8 can be more suitably made larger thanthe torsional spring constant of the first connection portions 5 and 6.

In the mirror device 1B, each of the wirings 14 and 16 is disposed topass through the first member 7 a. Since the wirings 14 and 16 aredisposed to pass through the first member 7 a which is subjected to asmaller stress than the second member 7 b during vibration of the secondmovable portion 4, deterioration of the wirings 14 and 16 in the secondconnection portion 7 can be suppressed.

In the mirror device 1A, the torsional spring constant of each of thesecond connection portions 7 and 8 around the X-axis is smaller than thetorsional spring constant of each of the first connection portions 5 and6 around the X. For this reason, the amplitude ratio can be more greatlysecured.

In the mirror device 1A, the total length of each of the secondconnection portions 7 and 8 is longer than the total length of each ofthe first connection portions 5 and 6. For this reason, the torsionalspring constant of the second connection portions 7 and 8 can besuitably made smaller than the torsional spring constant of the firstconnection portions 5 and 6.

In the mirror device 1A, each of the second connection portions 7 and 8extends in a meandering manner. For this reason, the torsional springconstant of the second connection portions 7 and 8 can be more suitablymade smaller than the torsional spring constant of the first connectionportions 5 and 6. Furthermore, since each of the second connectionportions 7 and 8 extends in a meandering manner, impact resistance canbe further improved, for example, as compared to when each of the secondconnection portions 7 and 8 extends linearly.

In the mirror devices 1A and 1B, the support portion 2, the firstmovable portion 3, the second movable portion 4, the first connectionportions 5 and 6, and the second connection portions 7 and 8 areconfigured by a semiconductor substrate. Even in the mirror devices 1Aand 1B configured in such a manner, a highly efficient and stableoperation can be implemented.

In the mirror device 1B, the portion of the wiring 16, the portion beinglocated on each of the first connection portions 5 and 6, is configuredby the diffusion region 41 where impurities are diffused in asemiconductor material. A relatively large stress is applied to theportion of the wiring 16 during swinging of the first movable portion 3,the portion being located on the first connection portions 5 and 6;however, since the portion is configured by the diffusion region 41,deterioration of the wiring in the first connection portions 5 and 6 canbe suppressed. Namely, when the portion is configured by the diffusionregion 41, metal fatigue, cracks, or the like can be further suppressed,and reliability can be further improved, for example, as compared towhen the portion is made of metal. Incidentally, since the wiring 16 isconnected to the coil 24 for electromotive force monitoring, arelatively smaller current flows through the wiring 16 than through, forexample, a wiring connected to a drive coil. For this reason, an extremereduction in resistance of the wiring 16 is not required.

In the minor device 1B, the width of the diffusion region 41 in thefirst movable portion 3 and the second movable portion 4 (the width W1of the end portion 41 a and the width W2 of the end portion 41 b) iswider than the width of the diffusion region 41 in the first connectionportions 5 and 6. For this reason, the resistance of the wiring 16 canbe reduced.

In the minor device 1B, the inner portion 16 a and the outer portion 16b of the wiring 16 are made of a metallic material, the inner portion 16a being located on the first movable portion 3 and the outer portion 16b being located on the second movable portion 4. Then, the width W4 ofthe contact portion between the diffusion region 41 and the innerportion 16 a and the width W5 of the contact portion between thediffusion region 41 and the outer portion 16 b are wider than the widthW3 of each of the first connection portions 5 and 6. For this reason,the resistance of the wiring 16 can be further reduced. In addition,even when a misalignment is generated during production or the like, thediffusion region 41 can be reliably electrically connected to each ofthe inner portion 16 a and the outer portion 16 b.

Modification Examples

The embodiments of the present disclosure have been described above;however, the present disclosure is not limited to the embodiments. Inthe mirror device 1A of the first embodiment, a drive force may beapplied to the second movable portion 4 to cause the first movableportion 3 to resonate. Namely, the structure of the mirror device 1A maybe used for outer drive. In this case, the first movable portion 3 maynot be provided with the coil 21, and the second movable portion 4 maybe provided with a coil for resonance operation of the first movableportion 3 in addition to being provided with the coil 22. Similarly tothe case of the mirror device 1B, in addition to a Lorentz forcegenerated by interaction between magnetic fields generated by the coilfor resonance operation and the magnetic field generation unit 9, theresonance of the first movable portion 3 at the resonance frequency canbe used to cause the mirror surface 10 to resonate around the X-axis.Incidentally, the coil for resonance operation may not be provided, andthe coil 22 may be used as a coil for the linear operation of the secondmovable portion 4 and the resonance operation of the first movableportion 3. Such a mirror device is designed and provided to satisfy onlyequation (23) but not to satisfy equation (22). As a result, the mirrorsurface 10 is operated at the natural angular frequency ω₂, so that theamplitude ratio can be 5 or more. In addition, the difference betweenthe natural angular frequencies ω₁ and ω₂ can be greatly secured, and anunnecessary resonance response can be suppressed to stabilize theoperation.

In the mirror device 1B of the second embodiment, a drive force may beapplied to the first movable portion 3 to cause the first movableportion 3 to resonate. Namely, the structure of the mirror device 1B maybe used for inner drive. In this case, the second movable portion 4 maynot be provided with the coil 22, and the first movable portion 3 may beprovided with a drive coil instead of or in addition to being providedwith the coil 24. Similarly to the case of the mirror device 1A, inaddition to a Lorentz force generated by interaction between magneticfields generated by the drive coil and the magnetic field generationunit 9, the resonance of the first movable portion 3 at the resonancefrequency can be used to cause the mirror surface 10 to resonate aroundthe X-axis. Such a mirror device is designed and provided to satisfyonly equation (32) but not to satisfy equation (33). As a result, themirror surface 10 is operated at the natural angular frequency ω₁, sothat the amplitude ratio can be 5 or more. In addition, the differencebetween the natural angular frequencies ω₁ and ω₂ can be greatlysecured, and an unnecessary resonance response can be suppressed tostabilize the operation.

In the mirror devices 1A and 1B, the mirror surface 10 is driven by anelectromagnetic force; however, the mirror surface 10 may be driven by apiezoelectric element or an electrostatic force. In this case, insteadof a coil, another drive element such as a piezoelectric film orelectrostatic comb teeth is provided. In the mirror device 1A, thesecond connection portions 7 and 8 may extend along the X-axis toconnect the second movable portion 4 and the support portion 2 to eachother on the X-axis. Even in this case, the first movable portion 3 isswingable around each of the X-axis and the Y-axis. In the mirror device1A, the second connection portions 7 and 8 may connect the secondmovable portion 4 and the support portion 2 to each other such that thesecond movable portion 4 is swingable around the second axisintersecting the first axis. Namely, the second axis may not necessarilybe orthogonal to the first axis. In the mirror device 1B, the secondmember 7 b may extend obliquely with respect to the X-axis. The secondmember 7 b may extend along the direction parallel to the Y-axis, toconnect the second movable portion 4 and the support portion 2 to eachother. In the above example, the lower limit value of the amplituderatio is set to 5; however, the lower limit value may be set to 10. Inthe mirror device 1B, the coil 24 may be used as a temperature detectioncoil (sensing coil).

In the mirror device 1A, a drive unit (the coil 21 or a drive element)which applies a drive force to the first movable portion 3 is providedin the first movable portion 3; however, the drive unit may be providedin the first movable portion 3 and/or the first connection portions 5and 6, for example, may be provided in each of the first connectionportions 5 and 6. In the mirror devices 1A and 1B, a drive unit (thecoil 22 or a drive element) which applies a drive force to the secondmovable portion 4 is provided in the second movable portion 4; however,the drive unit may be provided in the second movable portion 4 and/orthe second connection portions 7 and 8, for example, may be provided ineach of the second connection portions 7 and 8.

In the mirror devices 1A and 1B, the first movable portion 3 and/or thefirst connection portions 5 and 6 may be provided with a first driveelement (for example, coil) that directly applies a drive force to thefirst movable portion 3, and the second movable portion 4 and/or thesecond connection portions 7 and 8 may be provided with a second driveelement (for example, coil) that directly applies a drive force to thesecond movable portion 4. In such a configuration, the mirror surface 10(first movable portion 3) can be resonated around the X-axis by both ofthe drive force which is directly applied to the first movable portion 3by the first drive element and the drive force which is directly appliedto the first movable portion 3 by the vibration of the second movableportion 4 induced by the second drive element (by cooperation of both).When the mirror device 1A adopts the configuration, for example, thefirst movable portion 3 is provided with a coil that causes the firstmovable portion 3 to swing around the X-axis, and the second movableportion 4 is provided with a coil that causes the second movable portion4 to swing around the Y-axis, and a coil that vibrates the secondmovable portion 4 to cause the first movable portion 3 to swing aroundthe X. When the mirror device 1B adopts the configuration, for example,the first movable portion 3 is provided with a coil that causes thefirst movable portion 3 to swing around the X-axis, and the secondmovable portion 4 is provided with a coil that vibrates the secondmovable portion 4 to cause the first movable portion 3 to swing aroundthe X. Incidentally, when the mirror device 1B adopts the configuration,the mirror surface 10 (first movable portion 3) can be resonated aroundthe X-axis by the drive force which is directly applied to the firstmovable portion 3 by the first drive element, and the second movableportion 4 can be operated around the X-axis by the drive force which isdirectly applied to the second movable portion 4 by the second driveelement (namely, the configuration is also applicable to inner drive).

The materials and the shapes of the configurations are not limited tothe materials and the shapes described above, and various materials andshapes can be adopted. The first movable portion 3 and the secondmovable portion 4 may have any external shape such as a substantiallycircular shape, a substantially elliptical shape, a substantiallyquadrilateral shape, or a substantially diamond shape in plan view. Theframe 3 b may not be provided, and the first connection portions 5 and 6may be directly connected to the main body portion 3 a. At least one ofthe pair of widened portions 5 b may not be provided in the firstconnection portion 5, and at least one of the pair of widened portions 6b may not be provided in the first connection portion 6. The firstmovable portion 3 may have an optical surface other than the mirrorsurface 10, and may have, for example, a diffraction grating surface.Namely, an actuator device of the present disclosure may be a devicethat drives a surface other than the mirror surface 10.

In a mirror device 1D of a first modification example illustrated inFIG. 11, the first movable portion 3 swings around each of the X-axisand the Y-axis orthogonal to each other. The first movable portion 3 isformed, for example, in an elliptical shape in plan view. The firstconnection portions 5 and 6 extend linearly along the X-axis. The secondconnection portions 7 and 8 extend linearly along the Y-axis. The secondmovable portion 4 includes a frame-shaped portion 4 a surrounding thefirst movable portion 3, and an intermediate portion 4 b connected tothe frame-shaped portion 4 a.

The frame-shaped portion 4 a has a quadrilateral external shape in planview. The intermediate portion 4 b has, for example, a substantiallyquadrilateral external shape in plan view, and is formed in a frameshape. The intermediate portion 4 b includes a pair of first portions 4b 1, and a pair of second portions 4 b 2 connected to the pair of firstportions 4 b 1. The pair of first portions 4 b 1 extend along adirection parallel to the X-axis, and face each other in the Y-axisdirection. The pair of second portions 4 b 2 extend along a directionparallel to the Y-axis, and face each other in the X-axis direction. Thewidth (length in the Y-axis direction) of the first portion 4 b 1 iswider than the width (length in the X-axis direction) of the secondportion 4 b 2. The intermediate portion 4 b is disposed inside theframe-shaped portion 4 a, and is connected to the frame-shaped portion 4a in the first portion 4 b 1. The intermediate portion 4 b is connectedto the first connection portions 5 and 6 in the second portions 4 b 2.

In the mirror device 1D, a pair of piezoelectric films 51 are providedas drive units that applies a drive force to the first movable portion3. The pair of piezoelectric films 51 are disposed on the pair of firstportions 4 b 1 of the intermediate portion 4 b. Each of thepiezoelectric films 51 has, for example, a rectangular shape having along side along the direction parallel to the X-axis in plan view.

In the mirror device 1D, when a drive signal is input to thepiezoelectric films 51 via electrode pads and wirings (unillustrated),the piezoelectric films 51 are flexed in a longitudinal direction. Thetiming of deformation of the pair of piezoelectric films 51 iscontrolled, so that a drive force can be applied to the intermediateportion 4 b (second movable portion 4). For example, the pair ofpiezoelectric films 51 are alternately flexed. The rigidity of theintermediate portion 4 b is smaller than the rigidity of theframe-shaped portion 4 a, and the intermediate portion 4 b is moreflexible than the frame-shaped portion 4 a.

In the mirror device 1D, the drive of the first movable portion 3 isouter drive by which a drive force is applied to the second movableportion 4 to cause the first movable portion 3 to resonate. Even in themirror device 1D, similarly to the mirror device 1B, when the mirrorsurface 10 is operated at a first natural angular frequency ω₁, themirror device is designed and produced to satisfy equation (22), so thatthe amplitude ratio can be 5 or more, and when the mirror surface 10 isoperated at the natural angular frequency ω₂, the mirror device isdesigned and produced to satisfy equation (23), so that the amplituderatio can be 5 or more. In addition, the mirror device is designed andproduced to satisfy only one of equations (22) and (23) but not tosatisfy the other, so that the difference between the natural angularfrequencies ω₁ and ω₂ can be greatly secured, and an unnecessaryresonance response can be suppressed to stabilize the operation. Even inthe mirror device 1D, similarly to the mirror device 1B, a highlyefficient and stable operation can be implemented.

A mirror device 1E of a second modification example illustrated in FIG.12 differs from the mirror device 1D in that the piezoelectric film 51is disposed on the second portion 4 b 2 of the intermediate portion 4 b.Specifically, a pair of the piezoelectric films 51 are disposed side byside in the direction parallel to the Y-axis on a pair of the secondportions 4 b 2. Even in the mirror device 1E, the timing of deformationof each of the piezoelectric films 51 is controlled, so that a driveforce can be applied to the intermediate portion 4 b (second movableportion 4). Even in the mirror device 1E, similarly to the mirror device1B, a highly efficient and stable operation can be implemented.

REFERENCE SIGNS LIST

1A, 1B, 1C: mirror device (actuator device), 2: support portion, 3:first movable portion, 4: second movable portion, 5, 6: first connectionportion, 7, 8: second connection portion, 7 a: first member, 7 b: secondmember, 9: magnetic field generation unit (drive unit), 14, 15, 16, 17:wiring, 16 a: inner portion, 16 b: outer portion, 41: diffusion region,21: coil (drive unit), 22: coil (drive unit), 24: coil (electromotiveforce monitoring coil or sensing coil).

1: An actuator device comprising: a support portion; a first movableportion; a second movable portion that has a frame shape and surroundsthe first movable portion; a first connection portion that connects thefirst movable portion and the second movable portion to each other suchthat the first movable portion is swingable around a first axis; asecond connection portion that connects the second movable portion andthe support portion to each other such that the first movable portion isswingable around the first axis by vibrating the second movable portion;and a drive unit that applies a drive force to the second movableportion, wherein two natural angular frequencies ω₁ and ω₂ (where ω₁<ω₂)for vibration of the first movable portion and the second movableportion around the first axis satisfy one of the following firstequation (1) and second equation (2) and do not satisfy the other,$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 1} \right\rbrack & \; \\{0 < {1 - \left( \frac{\omega_{1}}{\omega_{ii}} \right)^{2}} \leq 0.2} & (1) \\\left\lbrack {{Equation}\mspace{14mu} 2} \right\rbrack & \; \\{0 < {\left( \frac{\omega_{2}}{\omega_{ii}} \right)^{2} - 1} \leq 0.2} & (2)\end{matrix}$ where in the above equations, ω_(ii)=(k_(i)/j_(i))^(1/2),k_(i) is a torsional spring constant of the first connection portionaround the first axis, and j_(i) is an inertia moment of the firstmovable portion around the first axis. 2: An actuator device comprising:a support portion; a first movable portion; a second movable portionthat has a frame shape and surrounds the first movable portion a firstconnection portion that connects the first movable portion and thesecond movable portion to each other such that the first movable portionis swingable around a first axis; a second connection portion thatconnects the second movable portion and the support portion to eachother; and a drive unit provided in at least one of the first movableportion and the first connection portion to apply a drive force to thefirst movable portion, wherein two natural angular frequencies ω₁ and ω₂(where ω₁<ω₂) for vibration of the first movable portion and the secondmovable portion around the first axis satisfy one of the following firstequation (3) and second equation (4) and do not satisfy the other,$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 3} \right\rbrack & \; \\{0 < \frac{4\omega_{io}^{2}}{\omega_{oo}^{2} - \omega_{1}^{2}} \leq 1} & (3) \\\left\lbrack {{Equation}\mspace{14mu} 4} \right\rbrack & \; \\{\frac{\omega_{oo}^{2} - \omega_{2}^{2}}{\omega_{io}^{2}} \leq {- 6}} & (4)\end{matrix}$ where in the above equations, ω_(io)=(k_(i)/j_(o))^(1/2),ω_(oo)=(k_(o)/j_(o))^(1/2), k_(i) is a torsional spring constant of thefirst connection portion around the first axis, k_(o) is a torsionalspring constant of the second connection portion around the first axis,and j_(o) is an inertia moment of the second movable portion around thefirst axis. 3: The actuator device according to claim 1, wherein thesecond connection portion connects the second movable portion and thesupport portion to each other such that the second movable portion isswingable around a second axis intersecting the first axis. 4: Theactuator device according to claim 1, wherein the two natural angularfrequencies satisfy the first equation and do not satisfy the secondequation, and a torsional spring constant of the second connectionportion around the first axis is larger than the torsional springconstant of the first connection portion around the first axis. 5: Theactuator device according to claim 4, wherein a total length of thesecond connection portion is shorter than a total length of the firstconnection portion. 6: The actuator device according to claim 4, whereinthe second connection portion includes a plurality of members, each ofwhich connects the second movable portion and the support portion toeach other, and the plurality of members are disposed side by side alonga direction intersecting the first axis. 7: The actuator deviceaccording to claim 6, wherein the plurality of members include a firstmember disposed on the first axis, and a pair of second members disposedto interpose the first member between the second members. 8: Theactuator device according to claim 7, further comprising: a wiringextending from the second movable portion to the support portion via thesecond connection portion, wherein the wiring is disposed to passthrough the first member. 9: The actuator device according to claim 1,wherein the two natural angular frequencies satisfy the second equationand do not satisfy the first equation, and a torsional spring constantof the second connection portion around the first axis is smaller thanthe torsional spring constant of the first connection portion around thefirst axis. 10: The actuator device according to claim 9, wherein atotal length of the second connection portion is longer than a totallength of the first connection portion. 11: The actuator deviceaccording to claim 9, wherein the second connection portion extends in ameandering manner. 12: The actuator device according to claim 1, whereinthe support portion, the first movable portion, the second movableportion, the first connection portion, and the second connection portionare configured by a semiconductor substrate. 13: The actuator deviceaccording to claim 1, further comprising: a sensing coil provided in thefirst movable portion; and a wiring that is connected to the sensingcoil and extends to the second movable portion via the first connectionportion, wherein the first connection portion is made of a semiconductormaterial, and a portion of the wiring, the portion being located on thefirst connection portion, is configured by a diffusion region whereimpurities are diffused in the semiconductor material. 14: The actuatordevice according to claim 13, wherein the diffusion region extends fromthe first connection portion to the first movable portion and the secondmovable portion, and a width of the diffusion region in the firstmovable portion and the second movable portion is wider than a width ofthe diffusion region in the first connection portion. 15: The actuatordevice according to claim 13, wherein a portion of the wiring, theportion being located on the first movable portion, and a portion of thewiring, the portion being located on the second movable portion, aremade of a metallic material, and a width of a contact portion betweenthe diffusion region and each of the portion located on the firstmovable portion and the portion located on the second movable portion iswider than a width of the first connection portion. 16: A method forproducing an actuator device including a support portion, a firstmovable portion, a second movable portion that has a frame shape andsurrounds the first movable portion, a first connection portion thatconnects the first movable portion and the second movable portion toeach other such that the first movable portion is swingable around afirst axis, a second connection portion that connects the second movableportion and the support portion to each other such that the firstmovable portion is swingable around the first axis by vibrating thesecond movable portion, and a drive unit that applies a drive force tothe second movable portion, wherein the actuator is produced such thattwo natural angular frequencies ω₁ and ω₂ (where ω₁<ω₂) for vibration ofthe first movable portion and the second movable portion around thefirst axis satisfy one of the following equation (5) and equation (6)and do not satisfy the other, $\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 5} \right\rbrack & \; \\{0 < {1 - \left( \frac{\omega_{1}}{\omega_{ii}} \right)^{2}} \leq 0.2} & (5) \\\left\lbrack {{Equation}\mspace{14mu} 6} \right\rbrack & \; \\{0 < {\left( \frac{\omega_{2}}{\omega_{ii}} \right)^{2} - 1} \leq 0.2} & (6)\end{matrix}$ where in the above equations, ω_(ii)=(k_(i)/j_(i))^(1/2),k_(i) is a torsional spring constant of the first connection portionaround the first axis, and j_(i) is an inertia moment of the firstmovable portion around the first axis. 17: A method for producing anactuator device including a support portion, a first movable portion, asecond movable portion that has a frame shape and surrounds the firstmovable portion, a first connection portion that connects the firstmovable portion and the second movable portion to each other such thatthe first movable portion is swingable around a first axis, a secondconnection portion that connects the second movable portion and thesupport portion to each other, and a drive unit that is provided in atleast one of the first movable portion and the first connection portionto apply a drive force to the first movable portion, wherein theactuator is produced such that two natural angular frequencies ω₁ and ω₂(where ω₁<ω₂) for vibration of the first movable portion and the secondmovable portion around the first axis satisfy one of the followingequation (7) and equation (8) and do not satisfy the other,$\begin{matrix}\left\lbrack {{Equation}\mspace{14mu} 7} \right\rbrack & \; \\{0 < \frac{4\omega_{io}^{2}}{\omega_{oo}^{2} - \omega_{1}^{2}} \leq 1} & (7) \\\left\lbrack {{Equation}\mspace{14mu} 8} \right\rbrack & \; \\{\frac{\omega_{oo}^{2} - \omega_{2}^{2}}{\omega_{io}^{2}} \leq {- 6}} & (8)\end{matrix}$ where in the above equations, ω_(io)=(k_(i)/j_(o))^(1/2),ω_(oo)=(k_(o)/j_(o))^(1/2), k_(i) is a torsional spring constant of thefirst connection portion around the first axis, k_(o) is a torsionalspring constant of the second connection portion around the first axis,and j_(o) is an inertia moment of the second movable portion around thefirst axis.